Given $ m \angle RPS = 5x + 24$, $ m \angle QPR = 9x + 16$, and $ m \angle QPS = 68$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Solution: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {9x + 16} + {5x + 24} = {68}$ Combine like terms: $ 14x + 40 = 68$ Subtract $40$ from both sides: $ 14x = 28$ Divide both sides by $14$ to find $x$ $ x = 2$ Substitute $2$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 5({2}) + 24$ Simplify: $ {m\angle RPS = 10 + 24}$ So ${m\angle RPS = 34}$.